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Theorem distrlem5pr 8651
Description: Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
distrlem5pr  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( A  .P.  B
)  +P.  ( A  .P.  C ) )  C_  ( A  .P.  ( B  +P.  C ) ) )

Proof of Theorem distrlem5pr
Dummy variables  x  y  z  w  v  u  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulclpr 8644 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
213adant3 975 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  B )  e. 
P. )
3 mulclpr 8644 . . . . 5  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C
)  e.  P. )
433adant2 974 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C )  e. 
P. )
5 df-plp 8607 . . . . 5  |-  +P.  =  ( x  e.  P. ,  y  e.  P.  |->  { f  |  E. g  e.  x  E. h  e.  y  f  =  ( g  +Q  h ) } )
6 addclnq 8569 . . . . 5  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
75, 6genpelv 8624 . . . 4  |-  ( ( ( A  .P.  B
)  e.  P.  /\  ( A  .P.  C )  e.  P. )  -> 
( w  e.  ( ( A  .P.  B
)  +P.  ( A  .P.  C ) )  <->  E. v  e.  ( A  .P.  B
) E. u  e.  ( A  .P.  C
) w  =  ( v  +Q  u ) ) )
82, 4, 7syl2anc 642 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
w  e.  ( ( A  .P.  B )  +P.  ( A  .P.  C ) )  <->  E. v  e.  ( A  .P.  B
) E. u  e.  ( A  .P.  C
) w  =  ( v  +Q  u ) ) )
9 df-mp 8608 . . . . . . . 8  |-  .P.  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. g  e.  w  E. h  e.  v  x  =  ( g  .Q  h ) } )
10 mulclnq 8571 . . . . . . . 8  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
119, 10genpelv 8624 . . . . . . 7  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( u  e.  ( A  .P.  C )  <->  E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z ) ) )
12113adant2 974 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
u  e.  ( A  .P.  C )  <->  E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z
) ) )
1312anbi2d 684 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( v  e.  ( A  .P.  B )  /\  u  e.  ( A  .P.  C ) )  <->  ( v  e.  ( A  .P.  B
)  /\  E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z
) ) ) )
14 df-mp 8608 . . . . . . . . 9  |-  .P.  =  ( w  e.  P. ,  v  e.  P.  |->  { f  |  E. g  e.  w  E. h  e.  v  f  =  ( g  .Q  h ) } )
1514, 10genpelv 8624 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( v  e.  ( A  .P.  B )  <->  E. x  e.  A  E. y  e.  B  v  =  ( x  .Q  y ) ) )
16153adant3 975 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
v  e.  ( A  .P.  B )  <->  E. x  e.  A  E. y  e.  B  v  =  ( x  .Q  y
) ) )
17 distrlem4pr 8650 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  A  /\  y  e.  B )  /\  (
f  e.  A  /\  z  e.  C )
) )  ->  (
( x  .Q  y
)  +Q  ( f  .Q  z ) )  e.  ( A  .P.  ( B  +P.  C ) ) )
18 oveq12 5867 . . . . . . . . . . . . . . . . . 18  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( v  +Q  u
)  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) )
1918eqeq2d 2294 . . . . . . . . . . . . . . . . 17  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( w  =  ( v  +Q  u )  <-> 
w  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) ) )
20 eleq1 2343 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( ( x  .Q  y )  +Q  ( f  .Q  z
) )  ->  (
w  e.  ( A  .P.  ( B  +P.  C ) )  <->  ( (
x  .Q  y )  +Q  ( f  .Q  z ) )  e.  ( A  .P.  ( B  +P.  C ) ) ) )
2119, 20syl6bi 219 . . . . . . . . . . . . . . . 16  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( w  =  ( v  +Q  u )  ->  ( w  e.  ( A  .P.  ( B  +P.  C ) )  <-> 
( ( x  .Q  y )  +Q  (
f  .Q  z ) )  e.  ( A  .P.  ( B  +P.  C ) ) ) ) )
2221imp 418 . . . . . . . . . . . . . . 15  |-  ( ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  /\  w  =  ( v  +Q  u
) )  ->  (
w  e.  ( A  .P.  ( B  +P.  C ) )  <->  ( (
x  .Q  y )  +Q  ( f  .Q  z ) )  e.  ( A  .P.  ( B  +P.  C ) ) ) )
2317, 22syl5ibrcom 213 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  A  /\  y  e.  B )  /\  (
f  e.  A  /\  z  e.  C )
) )  ->  (
( ( v  =  ( x  .Q  y
)  /\  u  =  ( f  .Q  z
) )  /\  w  =  ( v  +Q  u ) )  ->  w  e.  ( A  .P.  ( B  +P.  C
) ) ) )
2423exp4b 590 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( ( x  e.  A  /\  y  e.  B )  /\  (
f  e.  A  /\  z  e.  C )
)  ->  ( (
v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) ) )
2524com3l 75 . . . . . . . . . . . 12  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( f  e.  A  /\  z  e.  C ) )  -> 
( ( v  =  ( x  .Q  y
)  /\  u  =  ( f  .Q  z
) )  ->  (
( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) ) )
2625exp4b 590 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( ( f  e.  A  /\  z  e.  C )  ->  (
v  =  ( x  .Q  y )  -> 
( u  =  ( f  .Q  z )  ->  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  ->  (
w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C
) ) ) ) ) ) ) )
2726com23 72 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( v  =  ( x  .Q  y )  ->  ( ( f  e.  A  /\  z  e.  C )  ->  (
u  =  ( f  .Q  z )  -> 
( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( w  =  ( v  +Q  u
)  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) ) )
2827rexlimivv 2672 . . . . . . . . 9  |-  ( E. x  e.  A  E. y  e.  B  v  =  ( x  .Q  y )  ->  (
( f  e.  A  /\  z  e.  C
)  ->  ( u  =  ( f  .Q  z )  ->  (
( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) )
2928rexlimdvv 2673 . . . . . . . 8  |-  ( E. x  e.  A  E. y  e.  B  v  =  ( x  .Q  y )  ->  ( E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z )  ->  (
( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) ) )
3029com3r 73 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( E. x  e.  A  E. y  e.  B  v  =  ( x  .Q  y )  ->  ( E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z )  ->  (
w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C
) ) ) ) ) )
3116, 30sylbid 206 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
v  e.  ( A  .P.  B )  -> 
( E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z )  ->  ( w  =  ( v  +Q  u
)  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) ) )
3231imp3a 420 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( v  e.  ( A  .P.  B )  /\  E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z ) )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) )
3313, 32sylbid 206 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( v  e.  ( A  .P.  B )  /\  u  e.  ( A  .P.  C ) )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) )
3433rexlimdvv 2673 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( E. v  e.  ( A  .P.  B ) E. u  e.  ( A  .P.  C ) w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) )
358, 34sylbid 206 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
w  e.  ( ( A  .P.  B )  +P.  ( A  .P.  C ) )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) )
3635ssrdv 3185 1  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( A  .P.  B
)  +P.  ( A  .P.  C ) )  C_  ( A  .P.  ( B  +P.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152  (class class class)co 5858    +Q cplq 8477    .Q cmq 8478   P.cnp 8481    +P. cpp 8483    .P. cmp 8484
This theorem is referenced by:  distrpr  8652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-ni 8496  df-pli 8497  df-mi 8498  df-lti 8499  df-plpq 8532  df-mpq 8533  df-ltpq 8534  df-enq 8535  df-nq 8536  df-erq 8537  df-plq 8538  df-mq 8539  df-1nq 8540  df-rq 8541  df-ltnq 8542  df-np 8605  df-plp 8607  df-mp 8608
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